"In the next few days I will give an answer to the fans, who have always been at my side", said the five-time world player of the year, who appeared to target Real president Florentino Perez for a lack of support.
He started just one of Real's seven knockout matches in the Champions League.
"Who's disappointed? Perhaps they need to change the name of the Champions League to the CR7 Champions League". Madrid players were the first to approach their distraught opponent.
Bale's brilliant overhead kick had put Madrid 2-1 ahead as Klopp endured his sixth successive final defeat - a painful run that includes losing the 2013 Champions League final with Borussia Dortmund against Bayern Munich.
"We all know the result is 3-1 Real Madrid and that's why we are not in the best mood obviously".
Making an explosive impact as a second-half substitute, Bale showcased athleticism and technique of the highest quality by sending Madrid in front with only his fourth touch of the game in the 64th minute.
It may also ensure that Bale's future lies in the Spanish capital, particularly as Cristiano Ronaldo sensationally cast doubt on his future at the Bernabeu. The shock of the boys was obvious, we dropped a little bit too deep.
The Portugal superstar made his initial comments to beIN SPORTS and, when later pushed on the subject in the mixed zone, he conceded it had not been the ideal time to speak about his personal situation but reiterated he would reveal more soon.
Where Bale could go is less clear. Other destinations are unclear, though Manchester United has been reported in British media as a possible destination.
I am sure they did that back in the dressing room, once they got over their disappointment, and his manager will have a quiet word and support him too.
He has won four Champions Leagues, two LaLigas and two Copa del Reys during his career in Spain along with World Club titles.
However, his second mistake, when he fumbled Gareth Bale's shot over the line, was something I don't think he can recover from.